A linear filter approximation to the hammer/string interaction for use in a commuted synthesis piano model

In commuted synthesis of string instruments, the soundboard/body resonator is commuted to the excitation point and replaced by its own impulse response [Smith and Van Duyne, elsewhere in this session]. Hence, the highly nonlinear hammer/string interaction must be replaced by a commutable linear filter. Using the wave digital hammer computational model of the piano hammer [ 3300(A) (1994)], it was observed that the force pulse of a hammer striking an infinite string was qualitatively similar to the impulse response of a second-order filter with two real poles. Hence, good second- and higher-order filter designs based on physical data were possible. However, multiple humps may appear in the hammer force pulse on a terminated string due to returning string waves. It was observed that the magnitude spectra of the single hump spectrum and the multiple hump spectrum were similar in bandwidth, differing only in a slight ringing in the lower spectrum due to the lowpassed combing effect of the returning string waves. Therefore, an equalization filter was designed to summarize this combing effect by fitting a bank of parallel second-order sections to the complex ratio spectrum. Excellent linear piano hammer simulations were produced.